Seifert Matrix

Given a Seifert form f(x,y), choose a basis e_1, ..., e_(2g) for H_1(M^^) as a Z-module so every element is uniquely expressible as


with n_i integer. Then define the Seifert matrix V as the 2g×2g integer matrix with entries


For example, the right-hand trefoil knot has Seifert matrix

 V=[-1 1; 0 -1].

A Seifert matrix is not a knot invariant, but it can be used to distinguish between different Seifert surfaces for a given knot.

See also

Alexander Matrix

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Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200-203, 1976.

Referenced on Wolfram|Alpha

Seifert Matrix

Cite this as:

Weisstein, Eric W. "Seifert Matrix." From MathWorld--A Wolfram Web Resource.

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